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1、Longrangecorrelationsinbranchedpolymers.PiotrBialas?UniversiteitvanAmsterdam,InstituutvoorTheoretischeFysica,Valckenierstraat65,1018XEAmsterdam,TheNetherlandsAbstractWestudythecorrelationfunctionsinthebranchedpolymermodel.Althoughtherearenocorrelationsinthegrandcanonicalens
2、emble,whenlookingatthecanonicalensemblewe?ndnegativelongrangepowerlikecorrelations.Wepro-posethatasimilarmechanismexplainstheshapeofrecentlymeasuredcorrelationfunctionsintheelongatedphaseof4dsimplicialgravity[1].1IntroductionRecentlysomemeasurementsofthecurvature–curvaturec
3、orrelationfunctionsin4dsim-plicialgravitywerepresented[1].Thedataseemstoindicatenegativeandpowerlikebehaviorbothatthetransitionandintheelongatedphase.Thelatterseemssurprisingasonewouldexpecttheexponentialdecayofcorrelationsoutsidethecriticalregion.Thoseissuesarehardtostudyn
4、umericallyandevenharderanalytically.TogainsomeinsightintopossiblemechanismofthosecorrelationsweinvestigatehereasimplegeometricalarXiv:hep-lat/9511024v122Nov1995model:branchedpolymers(BP).Besidesitssimplicitytherearesomeothermotivationsforusingthismodel.TheBPdescribethec→∞li
5、mitof2dsimplicialgravity[2,3].ItisbelievedthatBPalsodescribetheelongatedphaseof4dsimplicialgravity[4].Howevertheexactcorrespondencebe-tweenthequantitiesmeasuredinref.[1]andBPisnotknown.NeverthelesswecanhopethattheBPmodelcapturesthegeneralfeaturesoftheelongatedphaseof4dsimpl
6、icialgravity.2ThemodelWeconsiderheretheensembleofplanarrootedplantedtrees.Atreeisagraphwithouttheclosedloops.Arootedtreeisatreewithonemarkedvertex(root)andplantedtreeisatreewhoseroot’sdegree(numberofbranches)isone.Twotreesareconsideredasdistinctiftheycannotbemappedoneachoth
7、erbyacontinuousdeformationoftheplane(see?g.1).Wedenotebynthetotalnumberofnon-rootverticesinthetreeandbyni?permanentaddress:InstituteofComp.Science,JagellonianUniversity,ul.Nawojki11,30-072Krak′ow,Poland1Figure1:Exampleofdistinctplantedrootedtrees=++...Figure2:Partitionfunct
8、ionthenumberofnon-rootverticesofthedegreei.Thenthe(grand)partitionfunctionisde?ned
